A probabilistic interval-based event calculus for activity recognition


Activity recognition refers to the detection of temporal combinations of ‘low-level’ or ‘short-term’ activities on sensor data. Various types of uncertainty exist in activity recognition systems and this often leads to erroneous detection. Typically, the frameworks aiming to handle uncertainty compute the probability of the occurrence of activities at each time-point. We extend this approach by defining the probability of a maximal interval and the credibility rate for such intervals. We then propose a linear-time algorithm for computing all probabilistic temporal intervals of a given dataset. We evaluate the proposed approach using a benchmark activity recognition dataset, and outline the conditions in which our approach outperforms time-point-based recognition.

This is a preview of subscription content, access via your institution.


  1. 1.

    Agrawal, J., Diao, Y., Gyllstrom, D., Immerman, N.: Efficient pattern matching over event streams. In: SIGMOD, pp. 147–160 (2008)

  2. 2.

    Akman, V., Selim, T., Erdoğan, J.L., Lifschitz, V., Turner, H.: Representing the Zoo World and the Traffic World in the language of the causal calculator. Artif. Intell. 153(1), 105–140 (2004)

    MATH  Article  Google Scholar 

  3. 3.

    Albanese, M., Chellappa, R., Cuntoor, N., Moscato, V., Picariello, A., Subrahmanian, V.S., Udrea, O.: PADS: A probabilistic activity detection framework for video data. IEEE Trans. Pattern Anal. Mach. Intell. 32(12), 2246–2261 (2010)

    Article  Google Scholar 

  4. 4.

    Alevizos, E., Skarlatidis, A., Artikis, A., Paliouras, G.: Probabilistic complex event recognition: A survey. ACM Comput. Surv. 50(5), 71:1–71:31 (2017)

    Article  Google Scholar 

  5. 5.

    Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)

    MATH  Article  Google Scholar 

  6. 6.

    Allison, L.: Longest biased interval and longest non-negative sum interval. Bioinformatics 19(10), 1294–1295 (2003)

    Article  Google Scholar 

  7. 7.

    Artikis, A., Sergot, M.J., Paliouras, G.: An event calculus for event recognition. IEEE Trans. Knowl. Data Eng. 27(4), 895–908 (2015)

    Article  Google Scholar 

  8. 8.

    Artikis, A., Skarlatidis, A., Portet, F., Paliouras, G.: Logic-based event recognition. Knowl. Eng. Rev. 27(4), 469–506 (2012)

    Article  Google Scholar 

  9. 9.

    Bacchus, F., Halpern, J.Y., Levesque, H.J.: Reasoning about noisy sensors in the situation calculus. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI), pp. 1933–1940. Morgan Kaufmann (1995)

  10. 10.

    Baral, C., Nam, T.H., Tuan, L.: Reasoning about actions in a probabilistic setting. In: Proceedings of the Eighteenth National Conference on Artificial Intelligence and Fourteenth Conference on Innovative Applications of Artificial Intelligence, pp. 507–512 (2002)

  11. 11.

    Brendel, W., Fern, A., Todorovic, S.: Probabilistic event logic for interval-based event recognition. In: The 24th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2011, Colorado Springs, CO, USA, 20-25 June 2011, pp. 3329–3336 (2011)

  12. 12.

    Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Comput. 35(8), 677–691 (1986)

    MATH  Article  Google Scholar 

  13. 13.

    Busany, N., Gal, A., Senderovich, A., Weidlich, M.: Interval-based queries over multiple streams with missing timestamps (2017)

  14. 14.

    Cervesato, I., Montanari, A.: A calculus of macro-events: Progress report. In: TIME, pp. 47–58 (2000)

  15. 15.

    Chesani, F., Mello, P., Montali, M., Torroni, P.: A logic-based, reactive calculus of events. Fundamenta Informaticae 105(1–2), 135–161 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Chittaro, L., Montanari, A.: Efficient temporal reasoning in the cached event calculus. Comput. Intell. 12(3), 359–382 (1996)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Chittaro, L., Montanari, A.: Temporal representation and reasoning in artificial intelligence: Issues and approaches. Ann. Math. Artif. Intell. 28(1), 47–106 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Clark, K.: Negation as failure. In: Gallaire, H., Minker, J. (eds.) Logic and Databases, pp. 293–322. Plenum Press (1978)

  19. 19.

    Craven, R.: Execution Mechanisms for the Action Language C+. Ph.D. thesis University of London (2006)

  20. 20.

    Cugola, G., Margara, A.: Processing flows of information: From data stream to complex event processing. ACM Comput. Surv. 44(3), 15:1–15:62 (2012)

    Article  Google Scholar 

  21. 21.

    D’Asaro, F.A., Bikakis, A., Dickens, L., Miller, R.: Foundations for a probabilistic event calculus. In: Balduccini, M., Janhunen, T. (eds.) Logic Programming and Nonmonotonic Reasoning - 14th International Conference, LPNMR 2017, Espoo, Finland, July 3-6, 2017, Proceedings, Lecture Notes in Computer Science, vol. 10377, pp. 57–63. Springer (2017)

  22. 22.

    D’Odorico, T.: An Ontological Analysis of Vague Motion Verbs, with an Application to Event Recognition. Ph.D. thesis, University of Leeds, UK (2013). http://etheses.whiterose.ac.uk/6909/

  23. 23.

    D’Odorico, T., Bennett, B.: Automated reasoning on vague concepts using formal ontologies, with an application to event detection on video data. In: Michael, L., Ortiz, C., Johnston, B. (eds.) Commonsense 2013—Proceedings of the 11th International Symposium on Logical Formalizations of Commonsense Reasoning (2013)

  24. 24.

    Doherty, P., Gustafsson, J., Karlsson, L., Kvarnström, J.: TAL: Temporal action logics language specification and tutorial. Electron. Trans. Artif. Intell. 2(3–4), 273–306 (1998)

    MathSciNet  Google Scholar 

  25. 25.

    Dousson, C., Maigat, P.L.: Chronicle recognition improvement using temporal focusing and hierarchization. In: IJCAI 2007, Proceedings of the 20th International Joint Conference on Artificial Intelligence, Hyderabad, India, January 6-12, 2007, pp. 324–329 (2007)

  26. 26.

    Dries, A., Kimmig, A., Meert, W., Renkens, J., den Broeck, G.V., Vlasselaer, J., Raedt, L.D.: Problog2: Probabilistic logic programming. In: Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2015, Porto, Portugal, September 7-11, 2015, Proceedings, Part III, pp. 312–315 (2015)

  27. 27.

    Eiter, T., Lukasiewicz, T.: Probabilistic reasoning about actions in nonmonotonic causal theories. In: Meek, C., Kjærulff, U. (eds.) UAI, pp. 192–199. Morgan Kaufmann (2003)

  28. 28.

    Fagin, R.: Combining fuzzy information from multiple systems. In: Proceedings of the Fifteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pp. 216–226. ACM Press (1996)

  29. 29.

    Gelfond, M., Lifschitz, V.: Representing action and change by logic programs. J. Log. Program. 17(2/3&4), 301–321 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Giatrakos, N., Alevizos, E., Artikis, A., Deligiannakis, A., Garofalakis, M.: Complex event recognition in the big data era. VLDB Journal (2019)

  31. 31.

    Gibson, J.: The ecological approach to visual perception (1979)

  32. 32.

    Ginsberg, M.L.: Multivalued logics: A uniform approach to reasoning in artificial intelligence. Comput. Intell. 4, 265–316 (1988)

    Article  Google Scholar 

  33. 33.

    Giunchiglia, E., Lee, J., Lifschitz, V., McCain, N., Turner, H.: Nonmonotonic causal theories. Artif. Intell. 153(1), 49–104 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Hajishirzi, H., Amir, E.: Sampling first order logical particles. In: Proceedings of the 24th Conference in Uncertainty in Artificial Intelligence (UAI), Helsinki, Finland, pp. 248–255. AUAI Press (2008)

  35. 35.

    Hölldobler, S., Karabaev, E., Skvortsova, O.: FLUCAP: A heuristic search planner for first-order MDPs. J. Artif. Intell. Res. (JAIR) 27(1), 419–439 (2006)

    MATH  Article  Google Scholar 

  36. 36.

    Iocchi, L., Lukasiewicz, T., Nardi, D., Rosati, R.: Reasoning about actions with sensing under qualitative and probabilistic uncertainty. ACM Trans. Comput. Log. 10(1), 5:1–5:41 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Kardas, K., Cicekli, N.K.: SVAS: surveillance video analysis system. Expert Syst. Appl. 89, 343–361 (2017)

    Article  Google Scholar 

  38. 38.

    van Kasteren, T., Englebienne, G., Kröse, B.: Human activity recognition from wireless sensor network data: Benchmark and software. In: Activity Recognition in Pervasive Intelligent Environments, Atlantis Ambient and Pervasive Intelligence. Atlantis Press (2010)

  39. 39.

    Katzouris, N., Artikis, A., Paliouras, G.: Online learning of event definitions. TPLP 16(5–6), 817–833 (2016)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Katzouris, N., Michelioudakis, E., Artikis, A., Paliouras, G.: Online learning of weighted relational rules for complex event recognition. In: Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD, pp. 396–413 (2018)

  41. 41.

    Kimmig, A., Demoen, B., Raedt, L.D., Costa, V.S., Rocha, R.: On the implementation of the probabilistic logic programming language ProbLog. Theory Pract. Logic Program. 11, 235–262 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Kowalski, R., Sadri, F.: Reconciling the event calculus with the situation calculus. J. Logic Program. 31(1), 39–58 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Kowalski, R.A., Sergot, M.J.: A logic-based calculus of events. Gen. Comput. 4(1), 67–95 (1986)

    MATH  Article  Google Scholar 

  44. 44.

    Kvarnström, J.: TALplanner and Other Extensions to Temporal Action Logic. Ph.D. thesis, Linköping (2005)

    Google Scholar 

  45. 45.

    List, T., Bins, J., Vazquez, J., Fisher, R.B.: Performance evaluating the evaluator. In: Proceedings 2nd Joint IEEE Int. Workshop on Visual Surveillance and Performance Evaluation of Tracking and Surveillance, pp. 129–136 (2005)

  46. 46.

    Mateus, P., Pacheco, A., Pinto, J., Sernadas, A., Sernadas, C.: Probabilistic situation calculus. Ann. Math. Artif. Intell. 32(1), 393–431 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    McCarthy, J., Hayes, P.J.: Some Philosophical Problems from the Standpoint of Artificial Intelligence. Stanford University (1968)

  48. 48.

    Michelioudakis, E., Artikis, A., Paliouras, G.: Semi-supervised online structure learning for composite event recognition. Mach. Learn. 108(7), 1085–1110 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Miller, R., Shanahan, M.: Some alternative formulations of the event calculus. In: Computational Logic: Logic Programming and Beyond, Essays in Honour of Robert A. Kowalski, Part II, Lecture Notes in Computer Science, pp. 452–490. Springer (2002)

  50. 50.

    Moldovan, B., Moreno, P., van Otterlo, M., Santos-Victor, J., De Raedt, L.: Learning relational affordance models for robots in multi-object manipulation tasks. In: Proceedings of International Conference on Robotics and Automation (ICRA), pp. 4373–4378 (2012)

  51. 51.

    Montali, M., Maggi, F.M., Chesani, F., Mello, P., van der Aalst, W.M.P.: Monitoring business constraints with the. Event Calculus ACM TIST, 5(1) (2014)

  52. 52.

    Morariu, V.I., Davis, L.S.: Multi-agent event recognition in structured scenarios. In: The 24th IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2011, Colorado Springs, CO, USA, 20-25 June 2011, pp. 3289–3296 (2011)

  53. 53.

    Mueller, E.T.: Commonsense Reasoning. Morgan Kaufmann (2006)

  54. 54.

    Mueller, E.T.: Event calculus and temporal action logics compared. Artif. Intell. 170(11), 1017–1029 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Paschke, A.: ECA-RuleML: An approach combining ECA rules with temporal interval-based KR event/action logics and transactional update logics. Tech. Rep. 11 Technische Universität München (2005)

  56. 56.

    Paschke, A., Bichler, M.: Knowledge representation concepts for automated SLA management. Decis. Support. Syst. 46(1), 187–205 (2008)

    Article  Google Scholar 

  57. 57.

    Paschke, A., Kozlenkov, A.: Rule-based event processing and reaction rules. In: Proceedings of the 3rd International Symposium on Rules (RuleML), Lecture Notes in Computer Science, vol. 5858, pp. 53–66. Springer (2009)

  58. 58.

    Pinto, J., Sernadas, A., Sernadas, C., Mateus, P.: Non-determinism and uncertainty in the situation calculus. Int. J. Uncert. Fuzziness Knowl.-Based Syst. 8(2), 127–149 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Pitsikalis, M., Artikis, A., Dreo, R., Ray, C., Camossi, E., Jousselme, A.: Composite event recognition for maritime monitoring. In: Proceedings of the 13th ACM International Conference on Distributed and Event-based Systems, DEBS, pp. 163–174 (2019)

  60. 60.

    Reiter, R.: Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems. MIT Press (2001)

  61. 61.

    Richardson, M., Domingos, P.: Markov logic networks. Mach. Learn. 62(1–2), 107–136 (2006)

    Article  Google Scholar 

  62. 62.

    Sadilek, A., Kautz, H.A.: Location-based reasoning about complex multi-agent behavior. J. Artif. Intell. Res. (JAIR) 43, 87–133 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Schiffel, S., Thielscher, M.: Reconciling situation calculus and fluent calculus. In: Proceedings of the National Conference on Artificial Intelligence, vol. 21, p. 287. AAAI Press (2006)

  64. 64.

    Selman, J., Amer, M.R., Fern, A., Todorovic, S.: PEL-CNF: Probabilistic event logic conjunctive normal form for video interpretation. In: ICCVW, pp. 680–687. IEEE (2011)

  65. 65.

    Shen, Z., Kawashima, H., Kitagawa, H.: Probabilistic event stream processing with lineage. In: Proc. of Data Engineering Workshop (2008)

  66. 66.

    Shet, V.D., Neumann, J., Ramesh, V., Davis, L.S.: Bilattice-based logical reasoning for human detection. In: (CVPR), pp. 1–8. IEEE Computer Society (2007)

  67. 67.

    Siskind, J.M.: Grounding the lexical semantics of verbs in visual perception using force dynamics and event logic. JAIR 15, 31–90 (2001)

    MATH  Article  Google Scholar 

  68. 68.

    Skarlatidis, A., Artikis, A., Filipou, J., Paliouras, G.: A probabilistic logic programming event calculus. TPLP 15(2), 213–245 (2015)

    MATH  Google Scholar 

  69. 69.

    Skarlatidis, A., Paliouras, G., Artikis, A., Vouros, G.A.: Probabilistic event calculus for event recognition. ACM Trans. Comput Logic, 16(2) (2015)

  70. 70.

    Thielscher, M.: From situation calculus to fluent calculus: State update axioms as a solution to the inferential frame problem. Artif. Intell. 111(1), 277–299 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  71. 71.

    Thielscher, M.: The qualification problem: A solution to the problem of anomalous models. Artif. Intell. 131(1), 1–37 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  72. 72.

    Van Belleghem, K., Denecker, M., De Schreye, D.: On the relation between situation calculus and event calculus. J. Logic Program. 31(1), 3–37 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  73. 73.

    Vouros, G.A., Vlachou, A., Santipantakis, G.M., Doulkeridis, C., Pelekis, N., Georgiou, H.V., Theodoridis, Y., Patroumpas, K., Alevizos, E., Artikis, A., Claramunt, C., Ray, C., Scarlatti, D., Fuchs, G., Andrienko, G.L., Andrienko, N.V., Mock, M., Camossi, E., Jousselme, A., Garcia, J.M.C.: Big data analytics for time critical mobility forecasting: Recent progress and research challenges. In: Proceedings of the 21th International Conference on Extending Database Technology, EDBT, pp. 612–623 (2018)

  74. 74.

    Wang, J., Domingos, P.: Hybrid Markov logic networks. In: Proceedings of the 23rd AAAI Conference on Artificial Intelligence, pp. 1106–1111. AAAI Press (2008)

  75. 75.

    Wang, Y.H., Cao, K., Zhang, X.M.: Complex event processing over distributed probabilistic event streams. Comput. Math. Appl. 66(10), 1808–1821 (2013)

    MATH  Article  Google Scholar 

  76. 76.

    Wu, E., Diao, Y., Rizvi, S.: High-performance complex event processing over streams. In: ACM SIGMOD, pp. 407–418 (2006)

  77. 77.

    Zhang, H., Diao, Y., Immerman, N.: Recognizing patterns in streams with imprecise timestamps. VLDB 3(1-2), 244–255 (2010)

    Google Scholar 

Download references


This work was supported by the INFORE project, which has received funding from the European Union’s Horizon 2020 research and innovation programme, under grant agreement No 825070. We would also like to thank Elias Alevizos for his feedback at the early stages of this work, and Evangelos Michelioudakis for his support in the experiments.

Author information



Corresponding author

Correspondence to Alexander Artikis.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Artikis, A., Makris, E. & Paliouras, G. A probabilistic interval-based event calculus for activity recognition. Ann Math Artif Intell 89, 29–52 (2021). https://doi.org/10.1007/s10472-019-09664-4

Download citation


  • Action languages
  • Complex event recognition
  • Probabilistic logic programming

Mathematics Subject Classification (2010)

  • 68T37